The US Ski Team is just a program of the USSA. It's not a whole different organization (more like a department).
Everyone who races in a World Cup race is a representative of some national association. Bode is a representative of the USSA and, from the narrow viewpoint of FIS rules and the like, he is a member of the US Ski Team.
What he doesn't do is accept support from the USSA (US Ski Team) or participate in training and other programs - the tradeoff is that he gets out of the obligations that the USSA expects racers to undertake in exchange for that support, so he can sleep where he wants, train where and when he wants, skip publicity events/sponsor meet-n-greets/autograph sessions with schoolkids etc. etc. ... at his own (not inconsiderable) expense.
The same has been true of a number of other racers. Most notably Kristina Koznick, for different reasons. Also, some racers with slightly less glorious careers. Kristen Mielke, I think, does or did train separately from the team. For that matter, the Nor-Am start lists are full of racers who are "representing" the US, but have scarcely ever even met a US Ski Team coach. There are from time to time racers who appear in World Cup Races, who may in some cases technically be "on" the US Ski Team, but maintain NCAA eligibility and race with their college teams.
FIS rules do, indeed, require that the national association's symbol appear on the front of the racers' helmets. They even specify the minimum size (9 square centimeters), and that it needs to be on whatever headgear you're wearing during flower ceremonies.